Discrete-time quantum walk with feed-forward quantum coin

A method for predicting essential proteins in heterogeneous networks based on discrete quantum walk

The finite dihedral group generated by one rotation and one reflection is the simplest case of the non-abelian group. Cayley graphs are diagrammatic counterparts of groups. In this paper, much attention is given to the Cayley graph of the dihedral group. Considering the characteristics of the elements in the dihedral group, we propose a model of three-state discrete-time quantum walk (DTQW) on the Caylay graph of the dihedral group with Grover coin. We derive analytic expressions for the position probability distribution and the long-time limit of the return probability starting from the origin. It is shown that the localization effect is governed by the size of the underlying dihedral group, coin operator and initial state. We also numerically investigate the properties of the proposed model via the probability distribution and the time-averaged probability at the designated position. The abundant phenomena of three-state Grover DTQW on the Caylay graph of the dihedral group can help the community to better understand and to develop new quantum algorithms.

We show that the Dirac quantum cellular automaton [Ann. Phys. 354 (2015) 244] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter, which plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions, and are similar to the ones achieved by highly localized states of the Dirac equation.

Chapter 5 - Discrete models

The cellular automaton (CA), a discrete model, is gaining popularity in simulations and scientific exploration across various domains, including cryptography, error-correcting codes, VLSI design and test pattern generation. This paper examines a two-state three-neighborhood nonuniform finite CA with a specified number of fixed points. We utilize a graph-based tool known as the next state rule minterm transition diagram (NSRTD) to analyze the spatiotemporal behavior of these cellular automata (CAs) across all lengths. We devise methods for synthesizing nonuniform single length cycle multi-attractor CAs (MACAs), a specialized class of irreversible CA with a predefined number of fixed points for any arbitrary length. Despite the method’s exponential worst-case time complexity, it offers the advantage of selecting rules for each CA cell from a set of candidate rules, ensuring the desired number of fixed points.

The operation of digital systems are based on time slotting and it finds many applications particularly in communication networks. It is important to analyze a digital network on some digital concepts. In this paper we analyze the network buffer through discrete time queueing system in which queue is discrete in nature and it is used to analyze the performance of the communication networks buffer. Discrete time models can be used to represents such type of systems in terms of slotted time units to models any physical system. More than one state changes in discrete time models from one time unit to another. Discrete time markov chains are used to describes the behavior of the network under the specific conditions and various performance measures are evaluated such as queue length, mean waiting time and mean number in the system.

Although cellular automata (CAs) have been invented only about three quarter of a century ago, much has been written about the fascinating space-time patterns these utter discrete dynamical systems can bring forth, which can be largely attributed to the fact that it is striking to notice that the dynamics of these overly simple dynamical systems can be so intriguing. Driven by the advances in the theory of dynamical systems of continuous dynamical systems, several quantitative measures have been proposed to grasp the stability of CAs, among which the Lyapunov exponent (LE) has received particular attention. Originally, the latter was understood as the Hamming distance between configurations evolved by the same CA from different initially close configurations, but it suffers from the important drawback that it can grow only linearly over time. In this paper, it will be shown how one should determine the LE of a two-state CA, in such a way that its meaning is in line with the framework of continuous dynamical systems. Besides, the proposed methodology will be exemplified for two CA families.KeywordsLyapunov ExponentCellular AutomatonInitial PerturbationPositive Lyapunov ExponentDefective CellThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we will look into a typical discrete dynamic model of common types of collective behaviors. Collective learning behavior of human group with interactions is first modelled according to the classical herding theory in psychology, and then the model is constructed and simulated based on the cellular automaton (CA) method. In our method, we assume that individuals learn to make their decisions repeatedly, and each individual's behavior at coming time step is driven by situation of system and by individual's anticipation with respect to future decisions of other individuals at current time step. CA is employed to compute and observe the visualized long-term behaviors of the system at higher group level, and a set of important statistical characters of the system are discussed. Our method is helpful for researches of modelling and simulating social dynamic systems and of researching higher-level behaviors of collective system with learning techniques.

As part of the comprehensive theme of the relationships between dynamic systems and computational theories, a very active area of research has been the relationships between the generic dynamic behavior of Cellular Automata (CA) and their computational abilities. Various investigations have been carried out on the computational power of CA, with concentrated efforts in the study of one-dimensional CA and their computational abilities. One of the approaches is the use of Genetic Algorithms (GA) to look for CA with a predefined computational behavior. A set of parameters which we have previously shown to be effective in helping forecast CA dynamic behavior, are used here as an auxiliary metric to guide the GA search. To this end, we modified selection, mutation and crossover of a GA, so as to incorporate the heuristic, and obtained very effective results in the evolution search for CA that can solve the so-called synchronization task.

Although deep learning has recently increased in popularity, it suffers from various problems including high computational complexity, energy greedy computation, and lack of scalability, to mention a few. In this paper, we investigate an alternative brain-inspired method for data analysis that circumvents the deep learning drawbacks by taking the actual dynamical behavior of biological neural networks into account. For this purpose, we develop a general framework for dynamical systems that can evolve and model a variety of substrates that possess computational capacity. Therefore, dynamical systems can be exploited in the reservoir computing paradigm, i.e., an untrained recurrent nonlinear network with a trained linear readout layer. Moreover, our general framework, called EvoDynamic, is based on an optimized deep neural network library. Hence, generalization and performance can be balanced. The EvoDynamic framework contains three kinds of dynamical systems already implemented, namely cellular automata, random Boolean networks, and echo state networks. The evolution of such systems towards a dynamical behavior, called criticality, is investigated because systems with such behavior may be better suited to do useful computation. The implemented dynamical systems are stochastic and their evolution with genetic algorithm mutates their update rules or network initialization. The obtained results are promising and demonstrate that criticality is achieved. In addition to the presented results, our framework can also be utilized to evolve the dynamical systems connectivity, update and learning rules to improve the quality of the reservoir used for solving computational tasks and physical substrate modeling.

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

In this paper, we study 2-dimensional nite cellular automata dened by hexagonal local rule with periodic boundary over the eld Z3.We construct the rule matrix corresponding to the hexagonal cellular automata.For some given coecients and the number of columns of hexagonal information matrix, we prove that the hexagonal cellular automata are reversible.

Complex patterns in a space–time discrete mathematical model of antibiotic resistance in hospitals with self-diffusion

Variables aggregation in a time discrete linear model

Effect of asynchronous updating on the stability of cellular automata